we're now going to talk about probably
the most famous formula in all of
finance and that's the black Scholes
formula sometimes called the black
Scholes Merton formula and it's named
after these gentlemen this right over
here is Fischer black this is Myron
Scholes and they really laid the
foundation for what led to the black
Scholes model and black Scholes formula
and that's why it has their name and
this is Bob Merton who really took what
black Scholes did and took it to another
level
to really get to our modern
interpretations of the black Scholes
model and the black Scholes formula and
all three of these gentlemen would have
won the Nobel Prize in Economics except
for the unfortunate fact that Fischer
black passed away before the award was
given but Myron Scholes and Bob Merton
did get the Nobel Prize for the work and
the reason why this is such a big deal
why it was his Nobel prize-worthy and
actually there's many reasons I could do
a whole series of videos on that is that
people had been trading stock option or
they've been trading options for a very
very very very long time they had been
trading them they had been buying them
they had been selling them it was a
major part of financial markets already
but there was no really good way of
putting our mathematical minds around
around how to value an option people had
a sense of the things that they cared
about and I would assume especially
options traders had a sense of the
things that they cared about when they
were trading options but we really
didn't have an analytical framework for
it and that's what the black Scholes
formula gave us but let's just before we
actually dive into this seemingly hairy
formula but the more we talk about it
hopefully it'll start to see him a lot
friendlier than it looks right now let's
start to get an intuition for the things
that we would care about if we were
thinking about the price of a stock
option so you would care about the stock
price you would care about the exercise
price you would actually especially care
about how much higher or lower the stock
price is relative to the exercise price
you would care about the risk-free
interest rate the risk-free interest
rate keep showing up when we think about
taking a present value of something if
we want to discount the value of
something back to today you would of
course think about well how long do I
have to actually exercise this
option and then finally this might look
a little bit bizarre at first but we'll
talk about in a second you would care
about how volatile that stock is and we
measure volatility as the standard
deviation of log returns for that
security so that's it seems very fancy
and we'll talk about that in more depth
in future videos but at just a intuitive
level just think about two stocks so
let's say that this is stock one right
over here and it jumps around and I'll
make them go flat just so we make no
judgment about whether it's a good
investment or so you have one stock that
kind of does that and then you have
another stock so let me actually draw
them on the same so let's say that is
stock one and then you have stock two
that does this that does this it jumps
around all over the place so this green
one right over here is stock two so you
could imagine stock two just in the way
we use the word volatile it is more
volatile it's a Wilder ride also if you
were looking at how dispersed the
returns are away from their mean you see
it has it has the the returns have more
dispersion it'll have a higher standard
deviation so stock two will have a
higher will have a higher volatility or
a higher standard deviation of
logarithmic returns and in a future
video we'll talk about why we care about
log returns and stock one would have a
lower volatility would have a lower
volatility so you can imagine options
are more valuable when you're dealing
with or if you're dealing with the stock
that has higher volatility that has
higher signal like this this feels like
it would drive the value of an option up
you would rather have an option when you
have something like this because look if
you're owning the stock man you have to
go off this is a wild ride but if you
have the option you don't you can ignore
the wildness and then it might actually
mate and then you can you can you can
exercise the option if it seems like the
right time to do it so it feels like if
you're just trading it that while the
most the more volatile something is the
more valuable and option would be on
that so with now that we've talked about
this let's actually look at the black
Scholes formula
and the variety that I have right over
here this is for a European call option
European call option we could do
something very similar for a European
put option so this right over here is a
European call option and remember
European call option it's it's
mathematically simpler than an American
call option and that there's only one
time which you can exercise it on the
exercise date on an American call option
you can exercise it at any point but
with that said let's try to at least
intuitively dissect the black-scholes
for me a little bit so the first thing
you have here you have this term that in
video involves the current stock price
and then you're multiplying it times
this function that's taking this as an
input and this is how we define that
input and then you have minus the
exercise price discounted back this
discounts back the exercise price times
that function again and now the input is
slightly different into that function
and just so that we have a little bit of
background about what this function n is
n is the cumulative distribution
function for a standard normal
distribution I know that seems might
seem a little bit daunting but you could
look at the statistics playlist and it
shouldn't be that bad this is
essentially saying for a standard normal
distribution the probability that your
random variable is less than or equal to
X and another way of thinking about that
if that sounds a little and it's all
explained in arched in our statistics
playlist if that was confusing but if
you want to think about a little bit
mathematically you also know that this
is going to be it's a probability it's
always going to be greater than 0 and it
is going to be less than 1 so with that
out of the way let's think about what
these pieces are telling us so this
right over here is dealing with the it's
the current stock of the price it's car
it's the current stock price and it's
being weighted by some type of a some
type of a a probability and so this is
essentially one way of thinking about in
very rough terms is this is what you're
going to get you're going to get the
stock and it's kind of being weighted by
the probability that you're actually
going to do this thing and I'm speaking
in very rough terms and then this term
right over here is what you pay this is
what you pay this is your exercise price
discounted back somewhat being weighted
and I'm speak
what's in I'm hand-waving a lot of the
mathematics but look what what's like
are we actually going to do this thing
are we actually going to exercise our
option so that makes sense right over
there and it makes sense if the stock
price is worth a lot more than the
exercise price and if we're definitely
going to do this let's say that d1 and
d2 were very very large numbers we're
definitely going to do this at some
point in time then it makes sense that
the value of the call option would be
the value of the would be the value of
the stock minus the exercise price
discounted back to today so this right
over here this is the discounting kind
of giving us the present value of the
Esk
exercise price and we have videos on
discounting and present value if you
find that a little bit daunting and it
also makes sense that the more the
higher the stock prices so we see that
right over here relative to the exercise
price the more that the option would be
worth but it also makes sense that the
higher the stock price relative to the
exercise price the more likely that we
will actually exercise the option and
you see that in both of these terms
right over here you have the ratio of
the stock price to the exercise price
ratio of the stock price to the exercise
price we're taking the natural log of it
but the higher this ratio is the larger
d1 or d2 is so that means the larger the
input into the cumulative distribution
function is which means the higher
probabilities we're going to get and so
it's it's a it's a higher chance we're
going to exercise this price and it
makes sense that then this is actually
going to have some value so that makes
sense the relationship between the stock
price and the exercise price the other
thing I will focus on because this tends
to be a deep focus of people who operate
with options is the volatility so we
already had an intuition that the higher
the volatility the higher the option
price so let's see where this factors
into this equation here so we don't see
it at this first level but it definitely
factors into d1 and d2 so in d1 the
higher your standard deviation of log
nor of your log returns so the higher
sigma we have a sigma in the numerator
and the denominator but in the numerator
we're squaring it so the higher sigma
will make d1 go up so Sigma Sigma goes
up D
one will go up let's think about what's
happening here well here we have a Sigma
it's still squared it's in the numerator
but we're subtracting it it's growing
boy this is going to grow faster than
this but we're subtracting it now so for
D to a higher Sigma is going to make D
to go down because we are subtracting it
so will this actually make can we
actually say that this is going to make
a higher Sigma is going to make the
value of our call option higher well
let's look at it if the value of our
Sigma goes up then d1 will go up then
this input this input goes up if that
input goes up our cumulative
distribution function of that input is
going to go up and so this term this
whole term is going to drive this whole
term up now what's going to happen here
well if d2 goes down then our cumulative
distribution function evaluated there is
going to go down and so this whole thing
is going to be lower and so we're going
to have to pay less so if we get more
and pay less than I'm speaking in very
hand wavy terms but this is just to
understand this isn't as intuitively
daunting as you might think but it looks
definitively that if the standard
deviation if the standard deviation of
our log returns or for volatility goes
up the value of our call option the
value of our European call option goes
up likewise using the same logic if our
volatility were to be lower then the
value of our call option would go down
I'll leave you there in future videos
we'll think about this in a little bit
more depth